Indented Angles.

Given two sheets of paper of the same size and form of a rectangle, fold them both in four equal parts, one lengthwise and the other sideways, as shown below.

Fig. 1.

When so folded take a fourth part off each. Part A in the figures. The question is now to cover quite exactly one of the remaining surfaces with the other, in cutting the latter in two perfectly equal parts.

Fig. 2.

To resolve this question take the surface (parts A having been detached), with which it is intended to cover the other, fold it again into equal parts, but this time in the opposite way to the one in which it was folded first, as indicated in Fig. 4: cut it out then in following the dotted line, F L, formed by the marks of the fold; this done one will obtain two parts absolutely equal, F L.

Fig. 3.

In order to cover the other surface, Fig. 3, all that is now necessary is to lower the angles, viz.: angle A’, must be in front of angle A, angle B’ in front of angle B, and angle C’ in front of angle C. When the angles are lowered in this way, the two surfaces will be quite similar, and can be covered one by the other.

Fig. 4.

This experiment can be made either with one or the other sheet, in lowering or raising the angles.

In the example shown here it is the fourth Figure which is destined to cover the other one.

When the operation is terminated as indicated above, part M, of Fig. 4, will be at M’ of Fig. 3, and part O at O’ of the covert figure.